Monitoring a sample containing a neutron source

ABSTRACT

The invention considers the frequency distributions of singles, doubles and triple neutron emission events from a sample under assay. The count rates are equated to mathematical functions related to the spontaneous fission rate, self-induced fission rate, detection efficiency and c,n rate with probability distribution assigned to each of those factors, the value of the product of all the probability distributions being increased to give an optimised solution and so provide a value of the spontaneous fission rate which is linked to the mass of the neutron source. The technique aims to provide increased accuracy and certainty compared with neutron coincidence counting based techniques.

This invention is concerned with improvements in and relating tomonitoring, in particular, but not exclusively, monitoring of nuclearmaterials.

Monitoring of nuclear waste, and plutonium waste in particular, isimportant both for accountancy purposes and in ensuring that criticalitysafety standards are maintained. It is important to maintain aninventory of the plutonium in a plant and during disposal, for instance,to know the exact amount of plutonium in a given volume of waste.

Determination of the mass of plutonium in a waste sample is presentlyconducted using neutron coincidence counting (NCC). On occasions it isnecessary to supplement these investigations with high resolution gammaspectrometry measurement to check the validity of assumptions on theisotopic make up of the sample. For waste samples such as drums, boxesand other handleable packages a detection chamber is normallyconstructed into which the samples to be measured are introduced.

The detection chamber principally consists of a series of neutrondetecting tubes placed in a polyethylene moderating material. Thesignals produced by the detectors are analysed electronically todetermine those signals attributable to spontaneous fissions of theplutonium rather than background radiation and other sources. Thespontaneous fissions of the plutonium produce 2, 3, 4 and higher numbersof neutrons for each event. The spontaneous fission rate is known to beproportional to the mass of certain isotopes and so to the neutronsource material as a whole. Signals detected within a certain timewindow are taken as being indicative of a pair, rather than a singleneutron, which can neither be attributed to the background or theplutonium with any certainty.

As the component of the signal relating to the pairs is indicative ofthe plutonium mass of certain isotopes, principally ²⁴⁰Pu then by makingcertain assumptions about the isotopic makeup of the sample the totalmass can be suggested.

As well as requiring such assumptions as to the material makeup whichare inappropriate given the likely variation over time of the isotopesin the waste, NCC also suffers from significant errors or inaccuraciesdue to the variable shielding effects of the different waste materialsin which the plutonium of interest may be distributed as well as due tothe different materials associated with the containers for the waste.For instance water, which will be present to varying degrees in thewaste, and polyethylene or PVC, which may be associated with thecontainer, are each strong neutron moderating materials. Variation inany of these can drastically effect the detection efficiency and hencethe result. Problems with the effects of changes in the detectionefficiency due to the materials actual position within the sample and asa consequence within the chamber are also encountered.

As an improvement it has been proposed to undertake multiplicitycounting (MC) in which the events producing single and triplet eventsare considered alongside the doublets. However, the count rates arisingrelate to four unknown parameters; the spontaneous fission rate(reflecting the isotopic make up of the material); the self-inducedfission are (known as multiplication); the detection efficiency; and theα,n reaction rate. Several other nuclear constants are also involved.The presence of three experimentally determinable factors, the single,double and triple count rates, still leaves the system unsolvable asthere are four unknowns. Detection of the quadlet counting rate isimpracticable due to the very limited number of detectable occurrencesfor such events. In even the best prior art systems, therefore, it isnecessary to set one of the unknowns at a predetermined value inreaching a solution.

Attempts to set any one of the self-induced fission rate; detectionefficiency; or α,n reaction rate are not accurate in waste samples.

The self-induced fission rate varies due to variations in materials andlocations with the result that even a small error has a substantialeffect on the end result due to the factors role in the equations. Thedetection efficiency differs from that obtained in calibration due tovariations in the location of the plutonium in the waste sample and dueto variations within the accompanying materials. The α,n rate changeswith the isotopic, chemical and materials in the waste matrix which willvary between samples. In each case errors in the end result ariseresulting in inefficient and costly waste disposal schemes or aninaccurate inventory.

According to a first aspect of the invention we provide a method ofmonitoring a sample containing a neutron source in which:—

-   -   i) signals from a plurality of neutron detectors are analysed        and the count rates for single, double and triple incidence of        neutrons on the detectors are determined;    -   ii) the single, double and triple count rates are equated to a        mathematical function related to the spontaneous fission rate,        self-induced fission rate, detection efficiency and α,n rate;    -   iii) a probability distribution is assigned to each of the        self-induced fission rate, detection efficiency and α,n reaction        rate and each of the counting rates to provide a probability        distribution factor for any given value;    -   iv) and the value of the product of all the probability        distribution factors is increased to give an optimised solution        and so provide a value for the spontaneous fission rate which is        linked to the mass of the neutron source.

In this way an accurate value for the mass of a neutron source within asample of material is provided without having to make hard assumptionsabout any of the variables within the system.

Quadruplet or high counts could be substituted for one of the single,pair of triple counts, but the number of events/unit time would bereduced accordingly.

The sample may be in the form of a piece of equipment, such as a glovebox, ventilation unit or the like; a component of a piece of equipment,such as a filter; or a container, such as a drum, box or other package.The container may be sealed. The sample may be of waste material, suchas material destined for disposal, or it may be material with a futureactive life for which an inventory is required.

The sample may include other materials besides the neutron source. Forinstance other materials such as non-neutron source elements, metals orcompounds, water, plastics, glass and other sealing materials may bepresent.

The neutron source may comprise one or more elements or compounds, oneor more isotopes of an element or mixtures of both. The neutron sourcemay be naturally occurring and/or arise from fission reaction products.Plutonium and ²⁴⁰Pu in particular are neutron sources which may requiresuch monitoring.

The neutron detectors may be of the ³He type. The detectors maybeprovided in polyethylene or other hydrogen providing material. In thisway the neutron source to be monitored can be controlled and theneutrons moderated to detectable energy levels. Alternative oradditional neutron absorbing materials, such as boron or cadmium, may beprovided to shield the detectors, for instance by positioning around thedetectors, against background events.

The detectors may be provided around all sides and most preferably aboveand/or below the sample during monitoring. Between 20 and 125 detectorsmay be provided. Preferably between 30 and 50 detectors are providedaround the sides of the sample. Preferably between 8 and 16 detectorsare provided above and below the sample.

Preferably each detector is provided with an amplifier (generally toincrease the pulse to a signal level suitable for further processing)and a discriminator (generally to reject or eliminate noise events.

The detector signals are preferably summed and fed to sequence analysingmeans. Preferably each pulse causes a time period to be considered, withother pulses being received in that period being associated with theinitial pulse. In this way sequences of single, double, triple andgreater numbers of neutron detections are obtained. Neutrons detectedwithin the time period are accepted as originating from the samespontaneous fission within the sample as the initial pulse. Preferablythe detections are subjected to a correction factor. The correctionfactor may account for accidental coincidences, for instance detectionsfrom different source simultaneous emissions which give the impressionof a pair or greater.

Preferably the time period lasts between 10 and 550 micro seconds andmost preferably between 50 and 100 micro seconds.

Preferably the start of the time period occurs between 5 and 10 microseconds after the initial pulse.

Preferably the singlet count rate is related to the spontaneous fissionrate, the self-multiplication factor, where$m = \frac{1 - p}{\left( {1 - p} \right)u_{I}}$and p=probability first neutron causes induced fission; the detectionefficiency and the α,n reaction rate by the functionR₁=ε.F_(S).M.v_(α1).(1+α)

Preferably the doublet counting rate is related to the spontaneousfission rate, the self-multiplication factor, where$m = \frac{1 - p}{\left( {1 - p} \right)u_{I}}$and p=probability first neutron causes induced fission; the detectionefficiency and the α,n reaction rate by the function$R_{2} = {ɛ^{2} \cdot F_{S} \cdot M^{2} \cdot v_{S2} \cdot \left( {1 + {\left( {M - 1} \right)\left( {1 + \alpha} \right)\frac{v_{S1}v_{l2}}{v_{S2}\left( {v_{l1} - 1} \right)}}} \right)}$

Preferably the triplet counting rate is related to the spontaneousfission rate, the self-multiplication factor, where$m = \frac{1 - p}{\left( {1 - p} \right)u_{I}}$and p=probability first neutron causes induced fission; the detectionefficiency and the α,n reaction rate by the function $\begin{matrix}{R_{3} = {ɛ^{3} \cdot F_{S} \cdot M^{3} \cdot v_{S3} \cdot \left( {1 + {2\left( {M - 1} \right)\frac{v_{S2}v_{l2}}{v_{S3}\left( {v_{l1} - 1} \right)}} +} \right.}} \\\left. {\left( {M - 1} \right)\left( {1 + \alpha} \right)\frac{v_{S1}v_{l3}}{v_{S3}\left( {v_{l1} - 1} \right)}\left( {1 + {2\left( {M - 1} \right)\frac{v_{l2}^{2}}{v_{l3}\left( {v_{S1} - 1} \right)}}} \right)} \right)\end{matrix}$

Preferably the probability distribution assigned to individual variablesor counting rates is a normal distribution or a flat distribution or atriangular distribution. Normal distributions are preferably used forone or more, and most preferably all, the counting rates. Triangulardistributions are preferably used for one or more, and most preferablyall, the individual variables, such as detector efficiency, fissionrate, multiplication distribution and alpha distribution. A flatdistribution is preferably used for the fission rate.

The probability distributions may be symmetrical. The probabilitydistributions may be skewed.

The distribution may be constrained within certain appliedconstraints/boundaries such that the probability distribution factor iszero beyond the constraints, particularly for triangular distributions.

The distribution may be modified such that the probability distributionfactor rapidly tends to zero beyond certain values, particularly fornormal distributions.

The constraints beyond which the distribution is zero or values beyondwhich a rapid transition to zero occurs, may be selected from one ormore of the following:—

-   -   for the detection efficiency 0≦E≦1;    -   for the self-induced fission rate M≧1;    -   for the α,n reaction α≧0.

The ranges of the distributions of one or more of the single count rate;the double count rate; and the triple count rate are preferably based onthe measured count rates and their standard deviation.

The lower and/or upper values of the distributions, may be providedaccording to one or more or all of the following ranges:—

-   -   a) for the self-induced fission rate: according to the        anticipated plutonium content; between 1.0 and 1.3, preferably        1.0 and 1.2; particularly for a high plutonium content (ie, >100        g Pu) between 1.0 and 1.2; particularly for a low plutonium        content (ie. <10 g Pu) between 1.00 and 1.01; and for        intermediate plutonium contents, intermediate ranges;    -   b) for the detection efficiency: according to the anticipated        moderator content; between 0 and 0.3; between 0 and 0.15 for        anticipated maximum efficiency levels; between 0.05 and 0.15 for        moderator containing materials;    -   c) for the alpha,n reaction rate: according to the form or forms        of plutonium anticipated; between 0 to 10, where plutonium        fluorides are anticipated as present; between 0 to 2, or 0 to 1        where plutonium metal is anticipated; greater than 0.0 where no        plutonium metal is anticipated, for instance between 0.4 and 1.0        or 0.5 and 1.0.

One or more of the constraints may be set according to informationgathered from a preceding isotopic consideration or analysis of thesample. Gamma spectrometry may be employed. Information about thepresence of plutonium metal and/or plutonium oxides and/or plutoniumfluorides and/or moderators and/or other species or components may beobtained and applied.

The permissible values for the spontaneous fission rate may beconstrained. Preferably the spontaneous fission rate is constrained toF≧0.

The distributions may be provided according to the general form:${{pdf}(ɛ)} = {\frac{1}{\left. \sqrt{}2 \right.\pi}\quad{\exp\left\lbrack {{{- 1}/2} \cdot \frac{\left( {ɛ - u_{ɛ}} \right)^{2}}{\sigma_{ɛ}}} \right\rbrack}}$

The initial set of values assigned to the variables may be selected bythe operator, predetermined or a function of prior samples which havebeen analysed.

The increasing, and preferably maximisation, of the product of theprobability distribution factors (pdf's) is preferably performed as aniterative process. The values of one, two, three and most preferably allfour of the variables are varied in each iteration. The variation orcorrection applied may be of a fixed value between iterations but avariable correction is preferred.

One or more of the probability distribution factors and/or thecorresponding level of the variable from an optimised or maximisedsolution may be used to redefine the range of the applicabledistribution used. One or more redefined distributions may be used in asubsequent optimisation or maximisation according to the method. Tightererrors apply as a result as tighter constraints would be applied. Theredefining of the distribution(s) and their use in optimisation may beperformed repeatedly.

The correction may be a correction vector. Most preferably thecorrection vector is derived from d(pfd)/d(x) where x is the variable,solved for zero.

In one method, because there are four variables, the par function may bepartially differentiated to yield a set of four equations:—

-   -   eg. μ_(ε)=0.12 σ_(E)=0.03    -   μm=1.02 σ_(m)=0.02    -   μ_(α)=0.4 σ_(α)=0.1

The product pdf may be partially differentiated to provide foursimultaneous equations. The equations may be${{\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial ɛ}{\Delta ɛ}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial F_{s}}\Delta\quad F_{s}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial M}\Delta\quad M} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial\alpha}\Delta\quad\alpha}} = {- \frac{\partial{pdf}}{\partial ɛ}}$${{\frac{\partial\left( \frac{\partial{pdf}}{\partial F_{s}} \right)}{\partial ɛ}{\Delta ɛ}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial F_{s}} \right)}{\partial F_{s}}\Delta\quad F_{s}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial F_{s}} \right)}{\partial M}\Delta\quad M} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial F_{s}} \right)}{\partial\alpha}\Delta\quad\alpha}} = {- \frac{\partial{pdf}}{\partial F_{s}}}$${{\frac{\partial\left( \frac{\partial{pdf}}{\partial M} \right)}{\partial ɛ}{\Delta ɛ}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial M} \right)}{\partial F_{s}}\Delta\quad F_{s}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial M} \right)}{\partial M}\Delta\quad M} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial M} \right)}{\partial\alpha}\Delta\quad\alpha}} = {- \frac{\partial{pdf}}{\partial M}}$${{\frac{\partial\left( \frac{\partial{pdf}}{\partial\alpha} \right)}{\partial ɛ}{\Delta ɛ}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial\alpha} \right)}{\partial F_{s}}\Delta\quad F_{s}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial\alpha} \right)}{\partial M}\Delta\quad M} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial\alpha} \right)}{\partial\alpha}\Delta\quad\alpha}} = {- \frac{\partial{pdf}}{\partial\alpha}}$

Alternatively or additionally the effects of small variations can bedetermined based on linearised equations in four dimensions. The use ofequations${{\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta ɛ}{\Delta ɛ}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\quad F_{s}}\Delta\quad F_{s}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad ɛ} \right)}{\delta\quad M}\Delta\quad M} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\alpha}\Delta\quad\alpha}} = {{{- \frac{\delta\quad{pdf}}{\delta ɛ}}{{\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta ɛ}{\Delta ɛ}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta\quad F_{s}}\Delta\quad F_{s}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta M}\Delta\quad M} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta\alpha}\Delta\quad\alpha}}} = {{{- \frac{\delta\quad{pdf}}{\delta\quad F_{s}}}{{\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta ɛ}{\Delta ɛ}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\quad F_{s}}\Delta\quad F_{s}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\quad M}\Delta\quad M} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\alpha}\Delta\quad\alpha}}} = {{{{- \frac{\delta\quad{pdf}}{\delta\quad M}}\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\alpha} \right)}{\delta ɛ}{\Delta ɛ}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\alpha} \right)}{\delta\quad F_{s}}\Delta\quad F_{s}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\alpha} \right)}{\delta\quad M}\Delta\quad M} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\alpha} \right)}{\delta\alpha}\Delta\quad\alpha}} = {- \frac{\delta\quad{pdf}}{\delta\alpha}}}}}$where${\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial F_{s}} \approx \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\quad F_{s}}} = \frac{\left( {\frac{\delta\quad{{pdf}\left( {ɛ,F_{s},M,\alpha} \right)}}{\delta ɛ} - \frac{\delta\quad{{pdf}\left( {ɛ,{F_{s} + {\Delta\quad F_{s}}},M,\alpha} \right)}}{\delta ɛ}} \right)}{\delta\quad F_{s}}$is the 2nd order derivative of${\left( \frac{\partial{pdf}}{\partial ɛ} \right) \approx \left( \frac{\partial{pdf}}{\partial ɛ} \right)} = {\frac{{{pdf}\left( {{ɛ + {\delta ɛ}},F_{s},M,\alpha,\mu_{R_{1}},{\mu_{R_{2}/}\mu_{R_{3}}}} \right)} - {{pdf}\left( {ɛ,F_{s},M,\alpha,\mu_{R_{1}},\mu_{R_{2}},\mu_{R_{3}}} \right)}}{\delta ɛ}\quad{{etc}.}}$may be employed. These functions may be solved to give a correctionvector, for instance $\quad\begin{bmatrix}\Delta_{ɛ} \\\Delta_{F} \\\Delta_{m} \\\Delta_{\alpha}\end{bmatrix}$

The method may handle the solution in matrix form.

Preferably the process is repeated until the correction vector is belowa given threshold. The threshold may be predetermined.

The initial pdf values for the allocated initial values are preferablyevaluated as positive ard negative correction vectors. Preferably eachsubsequent iterations pdf values are similarly evaluated. In this waythe effects of poor starting criteria are alleviated.

For a pdf the correction vector may be multiplied by a constant factorand divided by a constant factor and the corresponding new pdf comparedwith the preceding pdf, if the pdf value is greater than the precedingpdf the new correction vector is applied, if the pdf value is less thanthe preceding value the new correction factor is divided by the constantonce more and the new pdf for this further correction factor comparedwith the new preceding pdf, with the stages further being repeated ifnecessary. Preferably the multiplication factor is 32 and/or thedivisive factor is 0.5. Most preferably these steps are performed forboth positive and negative correction vectors.

If one or more of the initially allocated values corresponds to a pdf ofzero a new value may be allocated. Alternatively the count rate standarddeviations maybe multiplied by a constant factor, greater than 1. Inthis way a pdf value can be provided for which derivatives canbe-calculated. Preferably once the correction vector is below a certainthreshold the standard deviation can be divided by the initial constantor a lower figure and the solution maximised once more. This process ispreferably repeated until the inflationary factor is 1.

Preferably the method includes the provision of associated errorestimates. Total uncertainty within the system may be determined.

The associated error estimates may be determined by equations$\left( \sigma_{\hat{ɛ}} \right)^{2} = {\left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{{\hat{F}}_{s}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{\hat{M}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{M}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{M}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{M}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{\hat{\alpha}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$

Alternatively the associated error estimates may be obtained from theequations$\left( \sigma_{\hat{ɛ}} \right)^{2} = {\left( {\left( \frac{\delta\hat{ɛ}}{{\delta R}_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\delta\hat{ɛ}}{{\delta R}_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\delta\hat{ɛ}}{{\delta R}_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{{\hat{F}}_{s}}\text{)}^{2}} = {\left( {\left( \frac{\delta\hat{F_{s}}}{\delta\quad R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\delta\hat{F_{s}}}{{\delta R}_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\delta\hat{F_{s}}}{\delta\quad R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{\hat{M}}\text{)}^{2}} = {\left( {\left( \frac{\delta\hat{M}}{\delta\quad R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\delta\hat{M}}{\delta\quad R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\delta\hat{M}}{\delta\quad R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{\hat{\alpha}}\text{)}^{2}} = {\left( {\left( \frac{\delta\hat{\alpha}}{\delta\quad R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\delta\hat{\alpha}}{\delta\quad R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\delta\hat{\alpha}}{\delta\quad R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${{{where}\left( \frac{\partial\hat{ɛ}}{\partial R_{1}} \right)} \approx \left( \frac{\delta\hat{ɛ}}{\delta\quad R_{1}} \right)} = \frac{{\hat{ɛ}\left( {{\mu_{R_{1}} + {\delta\mu}_{R_{1}}},\mu_{R_{2}},\mu_{R_{3}}} \right)} - {\hat{ɛ}\left( {\mu_{R_{1}},\mu_{R_{2}},\mu_{R_{3}}} \right)}}{{\delta\mu}_{R_{1}}}$etc.where δμ_(R1)=a small change in μ_(R1): etc.ε (μ_(R1), μ_(R2), μ_(R3))=final estimate from solution to (μ_(R1),μ_(R2), μ_(R3)) count rates set.

According to a second aspect of the invention we provide:—

-   -   i) apparatus for monitoring a sample containing a neutron source        comprising one or more neutron detectors;    -   ii) processing means to receive signals from the detectors; and        wherein the signals are processed to provide a count rate value        indicative of the number of single, double and treble incidences        of neutrons on the detectors;    -   wherein the counting rates are mathematically related to the        spontaneous fission rate, self-induced fission rate, detection        efficiency and α,n reaction rate;    -   wherein a probability distribution is assigned by the processing        means to each of the self induced fission rate, detection        efficiency and α,n reaction rate and the three counting rates;        and    -   wherein the product of the probability distribution factors for        values of the variables is increased by the processing means,        the solution providing a value proportional to the mass of the        neutron source present.

The sample may be in the form of a piece of equipment, such as a glovebox, ventilation unit or the like; a component of a piece of equipment,such as a filter; or a container, such as a drum, box or other package.The container may be sealed. The sample may be of waste material, suchas material destined for disposal, or it may be material with a futureactive life for which an inventory is required.

The sample may include other materials besides the neutron source. Forinstance other materials such as non-neutron source elements, metals orcompounds, water, plastics, glass and other sealing materials may bepresent.

The neutron source may comprise one or more elements or compounds, oneor more isotopes of an element or mixtures of both. The neutron sourcemay be naturally occurring and/or arise from fission reaction products.Plutonium and ²⁴⁰Pu in particular are neutron sources which may requiresuch monitoring.

The neutron detectors may be of the ³He type. The detectors maybeprovided in polyethylene or other hydrogen providing material. In thisway the neutron source to be monitored can be controlled and theneutrons moderated to detectable energy levels. Alternative oradditional neutron shielding materials such as boron or cadmium may beprovided to shield the detectors, for instance by positioning around thedetectors.

The detectors may be provided around all sides and most preferably aboveand/or below the sample during monitoring. Between 20 and 125 detectorsmay be provided. Preferably between 30 and 50 detectors are providedaround the sides of the sample. Preferably between 6 and 16 detectorsare provided above and below the sample.

The detector signals are preferably summed and fed to sequence analysingmeans. Preferably each pulse causes a time period to be considered, withother pulses being received in that period being associated with theinitial pulse. In this way sequences of single, double, triple andgreater numbers of neutron detections are obtained. Neutrons detectedwithin the time period are accepted as originating from the samespontaneous fission within the sample as the initial pulse.

Other features and options available for the apparatus and its operatingprocedure are provide elsewhere within the application.

According to a third aspect of the invention we provide a samplemonitored using the method of the first aspect of the invention and/orusing the apparatus of the second aspect of the invention.

Various embodiments of the invention will now be described by way ofexample only and with reference to the accompanying drawings in which:—

FIG. 1 is an illustration of a detection chamber layout;

FIG. 2 illustrates a distribution assigned to the detection efficiency;

FIG. 3 a illustrates the probability distribution for a variable underconsideration;

FIG. 3 b shows the partial derivative function and a correction vectorfor the probability distribution of FIG. 4 a;

FIG. 4 a illustrates the probability distribution for an alternativedetermination;

FIG. 4 b shows the partial derivative function for the probabilitydistribution of FIG. 5 a illustrating the problem with poorly selectedinitial values;

FIG. 5 provides a comparison of NCC and multiplicity countingmeasurements of a ²⁵²Cf source in various matrix filled drums;

FIG. 6 illustrates a comparison of NCC and multiplicity countingmeasurements of a Pu source in various matrix filled drums;

FIG. 7 provides a comparison of NCC and multiplicity counting limits ofdetection for a drum containing 50 kg of PVC; and

FIG. 8 provides a comparison of NCC and multiplicity counting limits ofdetection for a drum containing 270 kg of steel hulls.

The detection chamber illustrated in FIG. 1 comprises a ten sidedchamber 2 with an openable wall portions 4 to provide access for a drum6 or the like to be monitored. Four ³He detectors are provided in eachside with twelve similar detectors in both the ceiling and floor 10 ofthe chamber.

Drums, 200 litre drums can be accommodated, may be lifted into thechamber, positioned there by forklift truck or conveyed on a bed ofrollers. Suspending the drum is preferred to avoid having to strengthenthe base of the chamber. Any such bearing surface could interfere withthe detection efficiency of the base detectors. Once positioned thechamber is shut.

The neutron detectors are of conventional type in tube configuration lmlong, 2.5 cm in diameter and operated at 4 atmosphere pressure and 960V.The outside of the chamber 2 is provided with an 24 cm thick layer ofpolyethylene to act as a neutron shield to any background radiation. Thedetectors are arranged vertically in bores in an inner layer of 8 cmthick polyethylene. A 1 mm layer of cadmium is provided on the insideand outside of the layer to prevent neutrons returning to the chamberand to prevent thermalise neutrons reaching the detectors.

Each detector is provided with a preamplifier, amplifier anddiscriminator to minimise dead time effects on the signals. The signalsare all summed and then fed to a frequency analyser.

The frequency analyser works on each pulse in the following manner. Areceived pulse opens after a preset period an observation interval withthe number of pulses falling within this period being counted. Afrequency distribution for the number of pulses detected within theobservation window is generated as a result. Singlets, doublets,triplets, quadruplets and higher sets can be expected with decreasingoccurrences.

The singlet, doublet and triplet counting rates are determined accordingto the following procedure.

The total neutron count rate observed during the measurement, N_(τ), isobtained simply by counting the number of signals. The first and secondfactorial moments of the frequency distribution, M_(m(1)) and N_(m(2)),are also calculated from the frequency table, using the followingequations:— $\begin{matrix}{N_{m{(1)}} = {\sum\limits_{x = 1}^{x = N_{\max}}{x \cdot N_{x}}}} & (1) \\{N_{m{(2)}} = {{\sum\limits_{x = 2}^{x = N_{\max}}{\begin{pmatrix}x \\2\end{pmatrix} \cdot N_{x}}} = {\sum\limits_{x = 2}^{x = N_{\max}}{\frac{x\left( {x - 1} \right)}{2} \cdot N_{x}}}}} & (2)\end{matrix}$where:

-   -   N_(x) is the normalised frequency to have x signals in the        observation interval.    -   N_(max) is the maximum number of signals observed in any        observation interval.

The total count rate and the first and second factorial moments are thenbackground corrected and used in the following equations to calculatethe singlet count rate, and the rate of correlated doublets andtriplets, R₁, R₂ and R₃ respectively. $\begin{matrix}{R_{1} = N_{T}} & (3) \\{R_{2} = \frac{{N_{T} \cdot N_{m{(1)}}} - {R_{1}^{2} \cdot \tau}}{f}} & (4) \\{R_{3} = \frac{{N_{T} \cdot N_{m{(2)}}} - {R_{2} \cdot R_{1} \cdot \tau \cdot \left\{ {f + W_{2}} \right\}} - {\frac{1}{2} \cdot R_{1}^{3} \cdot \tau^{2}}}{f^{2}}} & (5)\end{matrix}$where:

-   -   τ=length of the observation interval    -   l/λ=neutron die away time    -   T=pre-delay        f=e ^(−λT)(1−e ^(−λx))  (6) $\begin{matrix}        {W_{2} = {1 - {\frac{1}{\lambda\tau}\left( {1 - {\mathbb{e}}^{- {\lambda\tau}}} \right)}}} & (7)        \end{matrix}$

The counting rates obtained can then be applied to the followingequations which relate the counting rates to the spontaneous fissionrate, the self-induced fission rate, the detection efficiency and thealpha, n reaction rate. R₁ = ɛ ⋅ F_(s) ⋅ M ⋅ v_(s  1) ⋅ (1 + α)$R_{2} = {ɛ^{2} \cdot F_{s} \cdot M^{2} \cdot v_{s\quad 2} \cdot \left( {1 + {\left( {M - 1} \right)\left( {1 + \alpha} \right)\frac{v_{s\quad 1}v_{i\quad 2}}{v_{s\quad 2}\left( {v_{i\quad 1} - 1} \right)}}} \right)}$$R_{3} = {ɛ^{3} \cdot F_{s} \cdot M^{3} \cdot v_{s\quad 3} \cdot \left( {1 + {2\left( {M - 1} \right)\frac{v_{s\quad 2}v_{i\quad 2}}{v_{s\quad 3}\left( {v_{i\quad 1} - 1} \right)}} + {\left( {M - 1} \right)\left( {1 + \alpha} \right)\frac{v_{s\quad 1}v_{i\quad 3}}{v_{s\quad 3}\left( {v_{i\quad 1} - 1} \right)}\left( {1 + {2\left( {M - 1} \right)\frac{v_{i\quad 2}^{2}}{v_{i\quad 3}\left( {v_{s\quad 1} - 1} \right)}}} \right)}} \right)}$where ε=detector efficiency F_(s)=the spontaneous fission rate of thesample

-   -   M=the leakage multiplication    -   α=(α,n) to spontaneous fission ratio    -   n_(Sn)=the nth spontaneous fission factorial moment    -   n_(In)=the nth induced fission factorial moment    -   v_(Sn)=the nth spontaneous fission factorial moment (for        plutonium)    -   v_(In)=the nth induced fission factorial moment (for plutonium)

Rather than fixing one of the unknown values to obtain a solution to theequations the present invention seeks a best fit solution to theinformation obtained based on certain known absolute constraints andcertain probability distributions assigned to each of the unknowns.Information about the mean values and standard deviations for thecounting rate are determined experimentally. Information on thedistributions for the other parameters can be obtained by calibrationexperiments aswell as being based on experience and past tests.

The distributions are provided in the following manner:—

Detection Efficiency Distributionε˜N(μ_(e), σ_(e))0≧ε≧1where μ_(ε)=mean value of the efficiency distribution

-   -   σ_(ε)=standard deviation of the efficiency distribution        Fission Rate Distribution

Is constrained to be F_(s)≦0 but is otherwise left free floating as thevariable to be obtained.

Multiplication DistributionM˜N(μ_(M), σ_(M))M≧1where μ_(M)=mean value of the multiplication distribution

-   -   σ_(M)=standard deviation of the multiplication distribution        Alpha Distribution        α˜N(μ_(α), σ_(α))        α≧0        where μ_(α)=mean value of the alpha distribution    -   σ_(α)=standard deviation of the alpha distribution        Singles Count Rate Distribution        R₁˜N(μ_(R1), σ_(R1))        where μ_(R1)=mean value of the singles distribution obtained by        measurement    -   σ_(R1)=standard deviation of the singles distribution        Doubles Count Rate Distribution        R₂˜N(μ_(R2), σ_(R2))        where μ_(R2)=mean value of the doubles distribution obtained by        measurement    -   σ_(R2)=standard deviation of the doubles distribution        Triples Count Rate Distribution        R₃˜N(μ_(R3), σ_(R3))        where μ_(R3)=mean value of the triples distribution obtained by        measurement    -   σ_(R3)=standard deviation of the triples distribution

The probability distribution function (pdf) is assigned a normaldistribution with the pdf of any trial value being calculated by$z = \left( \frac{y_{i} - \mu_{y}}{\sigma_{y}} \right)$${{pdf}\left( y_{i} \right)} = {\frac{1}{\sqrt{2\pi}} \cdot {\exp\left( {{- 0.5}z^{2}} \right)}}$

FIG. 2 illustrates such a normal distribution for the detectionefficiency. Strictly speaking the pdf 20 should change abruptly to zerowhen either of the constraints for the efficiency is reached, i.e.greater than or equal to zero and less than or equal to one, however, ifsuch strict limits are applied an adverse effect on the solution processresults. As a consequence a constrained pdf 22 is provided but in a formwhich crosses the constraints to an extent.

The overall pdf for the variables can be determined by first calculatingthe count rates and then determining the product of all the individualpdf's. The solution is the set of values for the unknown parameterswhich give the maximum pdf product value, calculated according topdf(ε,F _(S) ,M,α,μ1_(R1),μ_(R2),μ_(R3))=pdf(ε)×pdf(F)×pdf(M)×pdf(α)×pdf(μ_(R1))×pdf(μ_(R2))×pdf(μ_(R3))

The method for determining the maximum solution to such a function is todifferentiate the function and solve for d(function)/d(parameter)=zero.As the pdf product has four input parameters it needs to be partiallydifferentiated to yield four simultaneous equations which can then besolved. Linearisation method or Taylor series method is used todetermine corrections (Δε; Δ F_(S1); ΔM; Δα) to be made to the trialvalue set selected in order to reduce the observed partial derivativesto zero.

FIG. 3 a shows a probability distribution 40 for a variable with thepartial derivative of this distribution being provided in FIG. 3 b. Thisillustrates in a single dimension the correction applied to an initialestimate 42 to give the next estimate 44. This in turn produces afurther correction factor and a subsequent further estimate 46 and soon.

In four dimensions the linearised equations are${{\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial ɛ}{\Delta ɛ}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial F_{s}}\Delta\quad F_{s}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial M}\Delta\quad M} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial\alpha}{\Delta\alpha}}} = {- \frac{\partial{pdf}}{\partial ɛ}}$${{\frac{\partial\left( \frac{\partial{pdf}}{\partial F_{s}} \right)}{\partial ɛ}{\Delta ɛ}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial F_{s}} \right)}{\partial F_{s}}\Delta\quad F_{s}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial F_{s}} \right)}{\partial M}\Delta\quad M} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial F_{s}} \right)}{\partial\alpha}{\Delta\alpha}}} = {- \frac{\partial{pdf}}{\partial F_{s}}}$$\begin{matrix}{{{\frac{\partial\left( \frac{\partial{pdf}}{\partial M} \right)}{\partial ɛ}{\Delta ɛ}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial M} \right)}{\partial F_{s}}\Delta\quad F_{s}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial M} \right)}{\partial M}\Delta\quad M} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial M} \right)}{\partial\alpha}{\Delta\alpha}}} = {- \frac{\partial{pdf}}{\partial M}}} \\{{{\frac{\partial\left( \frac{\partial{pdf}}{\partial\alpha} \right)}{\partial ɛ}{\Delta ɛ}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial\alpha} \right)}{\partial F_{s}}\Delta\quad F_{s}} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial\alpha} \right)}{\partial M}\Delta\quad M} + {\frac{\partial\left( \frac{\partial{pdf}}{\partial\alpha} \right)}{\partial\alpha}{\Delta\alpha}}} = {- \frac{\partial{pdf}}{\partial\alpha}}}\end{matrix}$

In order to simplify the analytical evaluation of these equations goodapproximations can be determined from the pdf product equation byobserving the effects of small changes in the individual parameters. Asan example involving a small change in the detection efficiency, ε, the1st order derivative${\left( \frac{\partial{pdf}}{\partial ɛ} \right) \approx \left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)} = \frac{\begin{matrix}{{{pdf}\left( {{ɛ + {\delta ɛ}},F_{s},M,\alpha,\mu_{R_{1}},\mu_{R_{2}},\mu_{R_{3}}} \right)} -} \\{{pdf}\left( {ɛ,F_{s},M,\alpha,\mu_{R_{1}},\mu_{R_{2}},\mu_{R_{3}}} \right)}\end{matrix}}{\delta ɛ}$is arrived at. This expands to give the second order derivativeapproximation${\frac{\partial\left( \frac{\partial{pdf}}{\partial ɛ} \right)}{\partial F_{s}} \approx \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\quad F_{s}}} = \frac{\begin{matrix}\left( {\frac{\delta\quad{{pdf}\left( {ɛ,F_{s},M,\alpha} \right)}}{\delta ɛ} -} \right. \\\left. \frac{\delta\quad{{pdf}\left( {ɛ,{F_{s} + {\Delta\quad F_{s}}},M,\alpha} \right)}}{\delta ɛ} \right)\end{matrix}}{\delta\quad F_{s}}$with the result that the linearisation equations are approximated by${{\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta ɛ}{\Delta ɛ}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\quad F_{s}}\Delta\quad F_{s}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\quad M}\Delta\quad M} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\alpha}{\Delta\alpha}}} = {- \frac{\delta\quad{pdf}}{\delta ɛ}}$${{\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta ɛ}{\Delta ɛ}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta\quad F_{s}}\Delta\quad F_{s}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta\quad M}\Delta\quad M} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta\alpha}{\Delta\alpha}}} = {- \frac{\delta\quad{pdf}}{\delta\quad F_{s}}}$${{\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta ɛ}{\Delta ɛ}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\quad F_{s}}\Delta\quad F_{s}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\quad M}\Delta\quad M} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\alpha}{\Delta\alpha}}} = {- \frac{\delta\quad{pdf}}{\delta\quad M}}$${{\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\alpha} \right)}{\delta ɛ}{\Delta ɛ}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\alpha} \right)}{\delta\quad F_{s}}\Delta\quad F_{s}} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\alpha} \right)}{\delta\quad M}\Delta\quad M} + {\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\alpha} \right)}{\delta\alpha}{\Delta\alpha}}} = {- \frac{\delta\quad{pdf}}{\delta\alpha}}$

These equations are most readily handled in matrix notation; D2×ΔX=D1where ${D\quad 1} = \begin{pmatrix}{- \frac{\delta\quad{pdf}}{\delta ɛ}} \\{- \frac{\delta\quad{pdf}}{\delta\quad F_{s}}} \\{- \frac{\delta\quad{pdf}}{\delta\quad M}} \\{- \frac{\delta\quad{pdf}}{\delta\alpha}}\end{pmatrix}$ ${D2} = \begin{pmatrix}\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta ɛ} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\quad F_{s}} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\quad M} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta ɛ} \right)}{\delta\alpha} \\\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta ɛ} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta\quad F_{s}} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta\quad M} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad F_{s}} \right)}{\delta\alpha} \\\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta ɛ} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\quad F_{s}} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\quad M} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad M} \right)}{\delta\alpha} \\\frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad\alpha} \right)}{\delta ɛ} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad\alpha} \right)}{\delta\quad F_{s}} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad\alpha} \right)}{\delta\quad M} & \frac{\delta\left( \frac{\delta\quad{pdf}}{\delta\quad\alpha} \right)}{\delta\alpha}\end{pmatrix}$ ${\Delta\quad X} = \begin{pmatrix}{\Delta\quad ɛ} \\{\Delta\quad F_{s}} \\{\Delta\quad M} \\{\Delta\quad\alpha}\end{pmatrix}$

Successive estimates are given by X_(n+1)=X_(n)+ΔX_(n) which process isrepeated until the correction vector is below a certain predeterminedthreshold which is accepted as negligible. A final solution$\begin{pmatrix}\hat{ɛ} \\\hat{F_{s}} \\\hat{M} \\\hat{\alpha}\end{pmatrix} = X_{final}$results. This solution provides a very good fit to the experimentalresults obtained. From the spontaneous fission rate value so obtainedthe mass of the neutron emitting isotopes in the sample can be obtained.This in turn can be linked to the overall neutron source mass.

In certain cases due to unusual sample conditions or due to poorselection of the starting criteria the correction vector provided may besuch that a true solution cannot be obtained. As illustrated in FIG. 4 awith the pdf 50 shown, the differentiated pdf, FIG. 5 b, may not changesign between the initial estimate 52 and the solution. If the initialestimate 54 is fortunately located then a true solution 56 will betended towards, but if the initial estimate 52 is not so fortunate afalse solution 58 will be tended towards.

To counter this a trial pdf is evaluated for both positive and negativecorrection factors.

The correction vector may substantially overestimate or underestimatethe distance to a solution, because the function is non-linear. Tocounter this the correction vector for a pdf value is multiplied by 32and then halved and the pdf value recalculated. This pdf value iscompared against the original. If the new pdf value is less than theprevious the process is repeated and the new value compared with the newprevious value. If the new value is greater than the previous then thecorrection vector is applied at that value. Otherwise the process isrepeated until such a situation is reached.

To avoid the situation where the initial pdf is zero and as aconsequence derivatives cannot be calculated, due to poorly selectedstarting conditions for instance, the count rate standard deviation isinflated in such a case. Multiplication by a large constant is employedto this end. After the process has converged towards a solution thevalue of this factor is reduced in stages and the process repeated untila solution is obtained with the inflationary factor set at zero.

In any experimentally derived result it is important to know the errorpossible in the result. This is particularly so for the monitoringsituations with which this method is principally concerned as theimplementation of the waste disposal based upon it must always act onthe worst possible case in meeting the critical safety ractors.

The associated error estimates are determined as follows. In each casethe precision of the final solution is dependant on the precision of thecount rates for the singlets, doublets and triplets. Assuming thesecount rates are independent the variance is given by$\left( \sigma_{\hat{ɛ}} \right)^{2} = {\left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{{\hat{F}}_{s}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{\hat{M}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{M}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{M}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{M}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{\hat{\alpha}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$

The partial derivatives again need not be determined but can beapproximated by${\left( \frac{\partial\hat{ɛ}}{\partial R_{1}} \right) \approx \left( \frac{\partial\hat{ɛ}}{\partial R_{1}} \right)} = \frac{\begin{matrix}{{\hat{ɛ}\left( {{\mu_{R_{1}} + {\delta\mu}_{R_{1}}},\mu_{R_{2}},\mu_{R_{3}}} \right)} -} \\{\hat{ɛ}\left( {\mu_{R_{1}},\mu_{R_{2}},\mu_{R_{3}}} \right)}\end{matrix}}{{\delta\mu}_{R_{1}}}$where δμ_(R1)=a small change in μ_(R1) etc.ε(μ_(R1), μ_(R2), μ_(R3))=final estimate from solution to (μ_(R1),μ_(R2), μ_(R3)) count rates set with the partial derivatives beingdetermined for each of the count rates eased on the observed rates andsmall deviations in each.

The approximate error estimates are thus given by$\left( \sigma_{\hat{ɛ}} \right)^{2} = {\left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{ɛ}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{{\hat{F}}_{s}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{F_{s}}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{\hat{M}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{M}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{M}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{M}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$${\text{(}\sigma_{\hat{\alpha}}\text{)}^{2}} = {\left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{1}} \right)\sigma_{R_{1}}} \right)^{2} + \left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{2}} \right)\sigma_{R_{2}}} \right)^{2} + \left( {\left( \frac{\partial\hat{\alpha}}{\partial R_{3}} \right)\sigma_{R_{3}}} \right)^{2}}$which in matrix notation becomes VX = DR × VR where${VX} = \begin{pmatrix}\left( \sigma_{\hat{ɛ}} \right)^{2} \\{\text{(}\sigma_{\hat{F_{s}}}\text{)}^{2}} \\\left( \sigma_{\hat{M}} \right)^{2} \\\left( \sigma_{\hat{\alpha}} \right)^{2}\end{pmatrix}$ ${VR} = \begin{pmatrix}\left( \sigma_{R_{1}} \right)^{2} \\{\text{(}\sigma_{R_{2}}\text{)}^{2}} \\\left( \sigma_{R_{3}} \right)^{2}\end{pmatrix}$ ${DR} = \begin{pmatrix}\left( \frac{\delta\hat{ɛ}}{\delta\quad R_{1}} \right)^{2} & \left( \frac{\delta\hat{ɛ}}{\delta\quad R_{2}} \right)^{2} & \left( \frac{\delta\hat{ɛ}}{\delta\quad R_{3}} \right)^{2} \\\left( \frac{\delta\hat{F_{s}}}{\delta\quad R_{1}} \right)^{2} & \left( \frac{\delta\hat{F_{s}}}{\delta\quad R_{2}} \right)^{2} & \left( \frac{\delta\hat{F_{s}}}{\delta\quad R_{3}} \right)^{2} \\\left( \frac{\delta\hat{M}}{\delta\quad R_{1}} \right)^{2} & \left( \frac{\delta\hat{M}}{\delta\quad R_{2}} \right)^{2} & \left( \frac{\delta\hat{M}}{\delta\quad R_{3}} \right)^{2} \\\left( \frac{\delta\hat{\alpha}}{\delta\quad R_{1}} \right)^{2} & \left( \frac{\delta\hat{\alpha}}{\delta\quad R_{2}} \right)^{2} & \left( \frac{\delta\hat{\alpha}}{\delta\quad R_{3}} \right)^{2}\end{pmatrix}$

A series of experimental measurements were conducted to test theperformance of the maximum-likelihood multiplicity analysis describedabove. These trials were carried out using a passive neutron countingtest rig containing 48³He neutron detector tubes encased in polythenemoderator. The chamber was sufficient to accommodate 200 litre drums andhad a neutron detecting efficiency of 15% for an empty drum and aneutron die-away-time of 600 m/s. A variety of matrix filled drums wereused in conjunction with Pu and ²⁵²Cf standard sources to simulate wastemeasurements.

In the first set of experimental trials, the reduction in systematicerrors (largely due to positional and matrix affects) achieved bymultiplicity counting compared with NCC using a ²⁵²Cf standard source tosimulate a large quantity of plutonium at various positions within wastematrix filled drums was employed. The specified probabilitydistributions for this analysis were (ε, ε₀, ε=(4%, 10%, 16%), (M, M₀,M₊)=1.00, 1.00, 1.20) and (α, α₀, α₊)=0.00, 0.00, 1.00). Thesedistributions were designed to represent a measurement scenario wherepossibly large amounts of plutonium were present within a matrix withhighly variable neutron moderating and/or absorption properties.

The results of these measurements (3000 s count times) are presented inTable 1. The first column of Table 1 gives details of the matrix and thesource position for each measurement. The second column shows themeasured net (i.e. background corrected) neutron singles, doubles andtriples count rates and their associated standard deviations. The thirdcolumn gives the results of a maximum-likelihood multiplicity analysisof the measured data. The fourth column shows the results that have beenobtained from a conventional NCC analysis in which the measured doublesrate is converted directly to a ²⁴⁰Pu_(eff) mass by means of acalibration factor. In this case, the calibration factor assumes ε=10%(i.e. the most likely value of the probability distribution specifiedfor the multiplicity analysis). TABLE 1 Net Count NCC Measurement Rates/Multiplicity Results Number ∈ −1 Results 240Pu_(eff)/g  #1 R1 = 22773+/−3 240Pu_(eff) = 132.7 g 100.8 g PVC (85 kg) R2 = 3471 +/−8 ∈ = 9.6%source @ B R3 = 269 +/−12 M = 1.00 α = 0.00  #2 R1 = 20311 +/−3240Pu_(eff) = 132.0 g  80.6 g PVC (85 kg) R2 = 2778 +/−8 ∈ = 8.6% source@A R3 = 185 +/−13 M = 1.00 α = 0.00  #3 R1 = 31861 +/−4 240Pu_(eff) =132.2 g 197.6 g Steel (270 kg) R2 = 6806 +/−12 ∈ = 13.5% source @ B R3 =686 +/−26 M = 1.00 α = 0.00  #4 R1 = 37947 +/−4 240Pu_(eff) = 132.1 g280.8 g Steel (270 kg) R2 = 9674 +/−19 ∈ = 16.0% source @ A R3 = 1315+/−39 M = 1.00 α = 0.00  #5 R1 = 28169 +/−3 240Pu_(eff) = 131.9 g 155.1g PVC (50 kg) R2 = 5344 +/−12 ∈ = 11.9% source @ B R3 = 484 +/−24 M =1.00 α = 0.00  #6 R1 = 33051 +/−5 240Pu_(eff) = 131.9 g 213.3 g PVC (50kg) R2 = 7348 +/−13 ∈ = 14.0% source @ A R3 = 831 +/−28 M = 1.00 α =0.00  #7 R1 = 28105 +/−5 240Pu_(eff) = 132.3 g 153.8 g Paper (20 kg) R2= 5304 +/−11 ∈ = 11.9% source @ B R3 = 511 +/−24 M = 1.00 α = 0.00  #8R1 = 35463 +/−39 240Pu_(eff) = 132.9 g 244.1 g Paper (20 kg) R2 = 8410+/−22 ∈ = 14.9% source @ A R3 = 1091 +/−52 M = 1.00 α = 0.00  #9 R1 =20604 +/−10 240Pu_(eff) = 132.1 g  82.9 g Water (105 kg) R2 = 2856 +/−9∈ = 8.7% source @ B R3 = 181 +/−13 M = 1.00 α = 0.00 #10 R1 = 13345 +/−2240Pu_(eff) = 131.8 g  34.8 g Water (105 kg) R2 = 1200 +/−5 ∈ = 5.7%source @ A R3 = 51 +/−6 M = 1.00 α = 0.00 #11 R1 = 29895 +/−4240Pu_(eff) = 131.7 g 174.6 g Polythene (22 kg) R2 = 6015 +/−11 ∈ =12.7% source @ B R3 = 569 +/−23 M = 1.00 α = 0.00 #12 R1 = 34554 +/−21240Pu_(eff) = 132.5 g 232.4 g Polythene (22 kg) R2 = 8004 +/−15 ∈ =14.6% source @ A R3 = 992 +/−34 M = 1.00 α = 0.0

The ²⁴⁰Pu_(eff) masses determined by the two techniques are comparedgraphically in FIG. 5. The arrow bars shown on the multiplicity resultshave be determined by the maximum-likelihood analysis and include boththe systematic and random components. The arrow bars shown on the MCresults also include both random and systematic components. Thesystematic term has been determined by assuming the same probabilitydistribution for the detection efficiencies were specified for themultiplicity analysis and the random term is simply the standarddeviation on the measured doubles rate.

The consistently accurate results obtained from the maximum-likelihoodMC analysis have been obtained by (correctly) varying ε (rather than Mand α) to match the predicted and measured count rates. This is trueeven in extreme cases such as the measurement number 10 where the sourcewas placed at the centre of a water-filled drum.

In a second set of tests a small Pu source (nominal ²⁴⁰Pu_(eff)mass=0.42 g) was measured under similar conditions to establish whatbenefits the maximum-likelihood MC analysis would provide for themeasurement of smaller quantities of Pu. The specified probabilitydistributions for the maximum likelihood MC analysis were (ε, ε₀,ε₊=(10%, 13%, 16%), (M, M₀, M₊)=1.00, 1.00, 1.01) and (α, α₀, α₊)=1.40,1.70, 2.00). The relatively tight distribution specified for the Mreflects the fact that for small quantities of Pu, it is safe to assumethat M will be close to unity. The relatively wide distributionspecified for αreflects the fact that this parameter will be difficultto estimate for real wastage of variations in chemical composition andimpurity content.

The results of these measurements (3000 s count times) are presented inTable 2 and a comparison of the ²⁴⁰Pu_(eff) mass is determined by themaximum-likelihood MC and NCC techniques are illustrated in FIG. 6.TABLE 2 Net NCC Measurement Count Rates/ Multiplicity Results Number ∈−1 Results 240Pu_(eff)/g #1 R1 = 124.34 +/−0.31 240Pu_(eff) = 0.40 g0.26 g PVC (85 kg) R2 = 3.88 +/−0.08 ∈ = 10.3% source @ A R3 = 0.32+/−0.05 M = 1.00 α = 1.93 #2 R1 = 125.93 +/−0.34 240Pu_(eff) = 0.36 g0.30 g Wood (48 kg) R2 = 4.50 +/−0.07 ∈ = 11.8% source @ B R3 = 0.29+/−0.03 M = 1.00 α = 1.89 #3 R1 = 156.94 +/−0.33 240Pu_(eff) = 0.41 g0.47 g Wood (48 kg) R2 = 7.10 +/−0.10 ∈ = 13.8% source @ A R3 = 0.69+/−0.10 M = 1.00 α = 1.70 #4 R1 = 121.46 +/−0.33 240Pu_(eff) = 0.38 g0.29 g PVC (50 kg) R2 = 4.41 +/−0.08 ∈ = 11.3% source @ B R3 = 0.48+/−0.14 M = 1.00 α = 1.76 #5 R1 = 159.82 +/−0.34 240Pu_(eff) = 0.38 g0.47 g PVC (50 kg) R2 = 7.09 +/−0.09 ∈ = 14.2% source @ A R3 = 0.61+/−0.04 M = 1.01 α = 1.92 #6 R1 = 137.75 +/−0.34 240Pu_(eff) = 0.37 g0.37 g Paper (20 kg) R2 = 5.62 +/−0.07 ∈ = 13.0% source @ B R3 = 0.47+/−0.05 M = 1.00 α = 1.79 #7 R1 = 173.61 +/−0.35 240Pu_(eff) = 0.39 g0.58 g Paper (20 kg) R2 = 8.77 +/−0.10 ∈ = 15.5% source @ A R3 = 0.86+/−0.06 M = 1.00 α = 1.83 #8 R1 = 133.44 +/−0.33 240Pu_(eff) = 0.40 g0.35 g Polythene R2 = 5.22 +/−0.09 ∈ = 12.0% (22 kg) source @ B R3 =0.66 +/−0.16 M = 1.00 α = 1.72 #9 R1 = 172.47 +/−0.36 240Pu_(eff) = 0.43g 0.54 g Polythene R2 = 8.09 +/−0.11 ∈ = 14.3% (22 kg) source @ A R3 =0.84 +/−0.09 M = 1.00 α = 1.73

As can be seen from FIG. 6, the maximum-likelihood MC analysis resultsare again consistently closer to the correct value than the conventionalNCC results. This emphasises the ability of the maximum-likelihoodtechnique to find the correct solution to the multiplicity equationswithout requiring any of the unknown parameters to be specified exactly.

The fact that the arrow bars shown for the MC analysis of the Pu sourceis significantly larger than those for the ²⁵²Cf measurements reflectsthe fact that the relative standard deviations on the measured countrates for the far smaller Pu source are significantly greater than thoseof the ²⁵²Cf source.

A third set of measurements were also made on drums filled with inactivematrices to evaluate the improvement in the limit of detection thatwould result from the use of the maximum-likelihood analysis techniquecompared with NCC. Several sets of hour long measurements were performedon the matrix filled drums and similar measurements were made with anempty chamber to determine the ambient background. The net count rateswere determined and then analysed by both maximum-likelihood MC andconventional NCC techniques. The specified probability distributions forthe maximum-likelihood MC analysis were (ε, ε₀, ε₊=(10%, 13%, 16%), (M,M₀, M₊)=1.00, 1.00, 1.01) and (α, α₀, α₊)=1.40, 1.70, 2.00).

The comparisons of the results obtained from the two techniques areshown graphically in FIG. 7 for a drum containing 50 kg of scrap PVC andFIG. 8 for a drum containing 270 kg of steel hulls. Also shown on thegraphs is an indication of the TRU/LLW segregation boundary at 100 nCi/g(calculated by assuming 94% ²³⁸Pu; 6% ²⁴⁰Pu).

The NCC results in FIG. 7 are scattered either side of zero because theydepend solely on the variation in the measured doubles rate from thematrix drum to the empty chain of measurements. The multiplicityresults, as discussed earlier, have been constrained to the physicallymeaningful situation of ²⁴⁰Pu_(eff) greater or equal to zero. Morespecifically, the magnitude and the measurement uncertainty on themultiplicity results is consistently less than half that of the NCCresults. This translates directly to a limited protection for themaximum-likelihood MC technique which is less than half that of aconventional NCC analysis performed under the same measurementconditions. This improvement over NCC detection limits is as expectedgiven that more useful information is incorporated into themaximum-likelihood MC analysis.

The results as shown in FIG. 8 for the drum containing the steel againshow reduced measurements uncertainties for the MC results compared withthe NCC results, but now show significant positive bias in the assayresults as evident despite the evidence of any spontaneous fissilematerial. Cosmic ray induced neutrons (which are generated primarily inhigh Z materials) are the likely cause of this affect. At its currentlevel the bias still allows re-categorisation against 100 nCi/g LLWlimit. This is despite the lack of background shielding on experimentalsystems without the use of matrix specific background measurements.

The improvement in limit of detection shown in FIGS. 6 and 7 indicatesthat the maximum likelihood multiplicity analysis technique will providea consistent and reliable method for the segregation of TRU/LLW at the100 nCi/g boundary.

The present invention therefore provides a technique whereby the unknownparameters for a waste sample can be determined with far greateraccuracy whilst avoiding undue assumptions about the system. In wasteinvestigations accurate Pu mass measurements can be made over the fullrange of wastes down to and below the 100 nCi/g LLW boundary. The resultis a more accurate and cost effective monitoring procedure withadvantages in inventory control and with significant cost savings inorganising waste disposal. The technique allows the low cost and highreliability associated with passive neutron counting systems to beretained. These advantages are obtained despite variations in themake-up of the waste and in the location of the neutron source withinthat waste. This improved monitoring is backed up by a firm indicationof the errors encountered. The parameter envelopes applied to thevarious functions can be updated as the monitoring of samplesprogresses. Thus the results from previous samples can improve themodelling of subsequent samples in the waste stream.

1-12. (canceled)
 13. A method of monitoring a sample containing aneutron source in which: i) signals from a plurality of neutrondetectors are analyzed and the count rates for single, double and tripleincidence of neutrons on the detectors are determined; ii) the single,double and triple count rates are equated to a mathematical functionrelated to the spontaneous fission rate, self-induced fission rate,detection efficiency and α,n reaction rate; iii) a probabilitydistribution is assigned to each of the spontaneous fission rate, theself-induced fission rate, detection efficiency and α,n reaction rateand each of the counting rates to provide a probability distributionfactor for any given value, wherein the probability distributionassigned to, the single, double, and triple count rates is a firstdistribution, the spontaneous fission rate is a second distribution, theself-induced fission rate is a third distribution, the detectorefficiency is a fourth distribution, the α,n reaction rate is a fifthdistribution; iv) and the value of the product of all the probabilitydistribution factors is increased to give an optimized solution and soprovide a value for the spontaneous fission rate which is linked to themass of the neutron source.
 14. The method of claim 13, wherein: thefirst distribution is a normal distribution; the second distribution isa flat distribution; the third distribution is triangular distribution;the fourth distribution is a triangular distribution; and the fifthdistribution is a triangular distribution.
 15. A method according toclaim 13 in which the signals comprise a series of pulses, each pulsecausing a time period to be considered, with other pulses being receivedin that period being associated with the initial pulse, the number ofpulses in the sequence giving the single, double, triple and greaternumbers of neutron counts.
 16. A method according to claim 13, whereinthe single neutron count rate (R₁) is related to the spontaneous fissionrate (F_(S)), the self induced fission rate (M), the detectionefficiency (ε) and the α,n reaction rate (α) by the function:R ₁=(ε)(F _(S))(M)(ν_(S1))(1+α), wherein ν_(S1) is a first spontaneousfission factorial moment for plutonium.
 17. A method according to claim13 in which the doublet counting rate R₂ is related to the spontaneousfission rate, the self-multiplication factor,$\left\lbrack {m = \frac{1 - p}{\left( {1 - p} \right)v_{I}}} \right\rbrack$the detection efficiency and the α,n reaction rate by the function$R_{2} = {ɛ^{2} \cdot F_{s} \cdot M^{2} \cdot {v_{s\quad 2}\left( {1 + {\left( {M - 1} \right)\left( {1 + \alpha} \right)\frac{v_{s\quad 1}v_{s\quad 2}}{v_{s\quad 2}\left( {v_{s\quad 2} - 1} \right)}}} \right)}}$where ν_(Sn) is the nth spontaneous fission factorial moment.
 18. Amethod according to claim 13 wherein the triplet counting rate R₃ isrelated to the spontaneous fission rate, the self-multiplication factor,$\left\lbrack {m = \frac{1 - p}{\left( {1 - p} \right)v_{I}}} \right\rbrack$the detection efficiency and the α,n reaction rate by the function$R_{3} = {ɛ^{3} \cdot F_{S} \cdot M^{3} \cdot v_{S3} \cdot \begin{pmatrix}{1 = {{2\left( {M - 1} \right)\frac{v_{S2}v_{S1}}{v_{S2}\left( {v_{S1} - 1} \right)}} =}} \\{\left( {M - 1} \right)\left( {1 + \alpha} \right)\frac{v_{S2}v_{S3}}{v_{S3}\left( {v_{S2} - 1} \right)}\left( {1 + {2\left( {M - 1} \right)\frac{v_{S1}^{2}}{v_{S3}\left( {v_{S1} - 1} \right)}}} \right)}\end{pmatrix}}$ where ν_(Sn) is the nth spontaneous fission factorialmoment.
 19. A method according to claim 13 in which the distribution(s)are constrained within certain applied constraints/boundaries, such thatthe probability distribution factor is zero beyond the constraints orsuch that the probability distribution factor tends to zero beyondcertain values.
 20. A method according to claim 13 in which one or moreof the constraints are set according to information gathered from apreceding isotopic consideration or analysis of the sample.
 21. A methodaccording to claim 13 in which the increasing, and, of the product ofthe probability distribution factors (pdf's) is performed as aniterative process.
 22. A method of monitoring a sample containing aneutron source having a neutron source mass, comprising: analyzingsignals from a plurality of neutron detectors; determining a singleincidence neutron count rate (R₁), a double incidence neutron count rate(R₂), and a triple incidence neutron count rate (R₃) associated with theneutron source based upon the analyzing; equating the single, double andtriple incidence neutron count rates to a mathematical function relatedto a spontaneous fission rate (F_(S)), a self-induced fission rate (M),a (α,n) reaction rate (α) and a detection efficiency (ε); assigning aprobability distribution to each of the spontaneous fission rate, selfinduced fission rate, the detection efficiency, the α,n reaction rateand each of the counting rates; obtaining probability distributionfactors for a set of trial values; calculating an overall value of aproduct of all the probability distribution factors; and varying one ormore of the trial values so as generate a maximal overall value for theproduct of all probability distribution factors, the value of thespontaneous fission rate being taken, wherein that value for thespontaneous fission rate is linked to the neutron source mass.
 23. Amethod as recited in claim 22, wherein the signals include a series ofpulses, comprising: receiving an initial pulse; after a preset period oftime after the initial pulse is received, opening an observationalinterval; and counting a number of pulses falling within theobservational interval, wherein the number of pulses is related to thesingle, double, triple, and greater numbers of neutron counts.
 24. Amethod according to claim 22 in which the probability distributionassigned to the spontaneous fission rate (F_(S)), the self inducedfission rate (M), the detection efficiency (ε) and the α,n reaction rate(α) is a normal distribution.
 25. A method according to claim 22 inwhich the probability distribution assigned to the spontaneous fissionrate (F_(S)), the self induced fission rate (M), the detectionefficiency (ε) and the α,n reaction rate (α) is a flat distribution. 26.A method according to claim 22 in which the probability distributionassigned to the spontaneous fission rate (F_(S)), the self inducedfission rate (M), the detection efficiency (ε) and the α,n reaction rate(α) is a triangular distribution.
 27. A method according to claim 24 inwhich a normal distribution is used for at least one of the countingrates.
 28. A method according to claim 25 in which a normal distributionis used for at least one of the counting rates.
 29. A method accordingto claim 26 in which a normal distribution is used for at least one ofthe counting rates.
 30. A method according to claim 24 in which a flatdistribution is used for at least one of the counting rates.
 31. Amethod according to claim 25 in which a flat distribution is used for atleast one of the counting rates.
 32. A method according to claim 26 inwhich a flat distribution is used for at least one of the countingrates.
 33. A method according to claim 24 in which a triangulardistribution is used for at least one of the counting rates.
 34. Amethod according to claim 25 in which a triangular distribution is usedfor at least one of the counting rates.
 35. A method according to claim26 in which a triangular distribution is used for at least one of thecounting rates.
 36. A method of monitoring a sample containing a neutronsource in which: i) signals from a plurality of neutron detectors areanalyzed and the count rates for single, double, and triple incidence ofneutrons on the detectors are determined; ii) the single, double, andtriple count rates are equated to a mathematical function related to thespontaneous fission rate, self induced fission rate, detectionefficiency and α,n reaction rate; iii) a probability distribution isassigned to each of the spontaneous fission rate, the self inducedfission rate, detection efficiency, and α,n reaction rate and each ofthe counting rates to provide a probability distribution factor for anygiven value; iv) and the value of the product of all the probabilitydistribution factors is increased to give an optimized solution and soprovide a value for the spontaneous fission rate which is linked to themass of the neutron source.
 37. A method according to claim 36 in whichthe signals comprise a series of pulses in a sequence, each pulsecausing a time period to be considered, with other pulses being receivedin that period being associated with the initial pulse, the number ofpulses in the sequence giving the single, double, triple, and greaternumber of neutron counts.
 38. A method according to claim 36 in whichthe probability distribution assigned to individual variables orcounting rates is a normal distribution or a flat distribution or atriangular distribution.
 39. A method according to claim 36 in which anormal distribution is used for one or more, the-counting rates.
 40. Amethod according to claim 36 in which triangular distributions are usedfor one or more of the individual variables, such as detectorefficiency, fission rate, multiplication distribution and alphadistribution.
 41. A method according to claim 36 in which a flatdistribution is used for the fission rate.